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Periodic solutions of parabolic systems with nonlinear boundary conditions. (English) Zbl 0932.35111

The existence and the stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions is investigated. The method of upper and lower solutions and its associated monotone iterations is used. This method implies the existence of maximal and minimal periodic solutions, which can be computed from a linear iteration process in the same way as for parabolic initial-boundary value problems. A sufficient condition for the stability of a periodic solution is also given. These results are applied to models arising in chemical kinetics, ecology, and population biology.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B10 Periodic solutions to PDEs
35K55 Nonlinear parabolic equations
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[1] Ahmad, S.; Lazer, A. C., Asymptotic behavior of solutions of periodic competition diffusion systems, Nonlinear Anal., 13, 263-283 (1989)
[2] Amman, H., Periodic solutions of semilinear parabolic equations, Nonlinear Analysis (1978), Academic Press: Academic Press New York, p. 1-29
[3] Bange, D. W., Periodic solutions of a quasilinear parabolic differential equation, J. Differential Equations, 17, 61-72 (1975) · Zbl 0291.35051
[4] Brown, K. J.; Hess, P., Positive periodic solutions of predator-prey reaction-diffusion systems, Nonlinear Anal., 16, 1147-1158 (1991) · Zbl 0743.35030
[5] Du, Y., Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 19, 1043-1066 (1996) · Zbl 0858.35057
[6] Feng, W.; Lu, X., Asymptotic periodicity in logistic equations with discrete delays, Nonlinear Anal., 26, 171-178 (1996) · Zbl 0842.35129
[7] Five, P., Solutions of parabolic boundary problems existing for all times, Arch. Rational Mech. Anal., 16, 155-186 (1964) · Zbl 0173.38204
[8] Friedman, A., Partial Differential Equations of Parabolic type (1964), Prentice-Hall: Prentice-Hall Engelwood Cliffs · Zbl 0144.34903
[9] Fu, S.; Ma, R., Existence of a global coexistence state for periodic competition diffusion systems, Nonlinear Anal., 28, 1265-1271 (1997) · Zbl 0871.35051
[10] Hess, P., Periodic-Parabolic Boundary Value Problems and Positivity. Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., 247 (1991), Longman Scientific and Technical: Longman Scientific and Technical New York · Zbl 0731.35050
[11] Kolesov, Ju. S., Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc., 21, 114-146 (1970) · Zbl 0226.35040
[12] Kusano, T., Periodic solutions of the first boundary value problem for quasilinear parabolic equations of second order, Funckcial. Ekvac., 9, 129-137 (1966) · Zbl 0154.36101
[13] Leung, A. W.; Ortega, L. A., Existence and monotone scheme for Time-periodic nonquasimonotone reaction-diffusion systems: Application to autocatalytic chemistry, J. Math. Anal. Appl., 221, 712-733 (1998) · Zbl 0914.35145
[14] Liu, B. P.; Pao, C. V., Periodic solutions of coupled semilinear parabolic boundary value problems, Nonlinear Anal., 6, 237-252 (1982) · Zbl 0499.35012
[15] Lu, X.; Feng, W., Periodic solution and oscillation in a competition model with diffusion and distributed delay effect, Nonlinear Anal., 27, 699-709 (1996) · Zbl 0862.35134
[16] Nakao, M., On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19, 371-385 (1975) · Zbl 0328.35050
[17] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044
[18] Pao, C. V., System of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl., 205, 157-185 (1997) · Zbl 0880.35126
[19] Pao, C. V., Numerical analysis of coupled system of nonlinear parabolic equations, SIAM J. Numer. Anal., 36, 393-416 (1999) · Zbl 0921.65061
[20] Tineo, A., Existence of global coexistence for periodic competition diffusion systems, Nonlinear Anal., 19, 335-344 (1992) · Zbl 0779.35058
[21] Tineo, A., Asymptotic behavior of solutions of a periodic reaction-diffusion system of a competitor-competitor-mutualist model, J. Differential Equations, 108, 326-341 (1994) · Zbl 0806.35095
[22] Tsai, L. Y., Periodic solutions of nonlinear parabolic differential equations, Bull. Inst. Math. Acad. Sinica, 5, 219-247 (1977) · Zbl 0375.35031
[23] Tyson, J. J., The Belousov-Zhabotinskii Reaction. The Belousov-Zhabotinskii Reaction, Lecture Notes in Biomath., 10 (1976), Springer: Springer New York · Zbl 0342.92001
[24] Varga, R. S., Matrix Iterative Analysis (1962), Prentice Hall: Prentice Hall Engelwood Cliffs · Zbl 0133.08602
[25] Zhao, X. Q., Global asymptotic behavior in a periodic competitor-competitor-mutualist system, Nonlinear Anal., 29, 551-568 (1997) · Zbl 0876.35058
[26] Zheng, S., A reaction-diffusion system of a competitor-competitor-mutualist model, J. Math. Anal. Appl., 124, 254-280 (1987) · Zbl 0658.35053
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